UC Davis

Over the past century, mathematical epidemiology has grown to be one of the triumphs of applied mathematics and mathematical biology. It has drawn influence and insight from a variety of related fields, including mathematics, physics, chemistry, biology, ecology, and social science, among others. With tools ranging from simple ordinary differential equation models to highly complex stochastic simulations, mathematical models of epidemic spread have had significant theoretical and practical impacts. In the past two decades, the development of network science as a discipline has lead to a new modeling paradigm in mathematical epidemiology. Networks can capture aspects of social structure that are critical to disease spread, allowing for models that balance parsimony and complexity.

In this dissertation, I consider questions of model construction, analysis, and application that are united under the framework of modeling epidemics on networks. In Chapter 2, we consider an existing low-dimensional model of an SIS disease on a network. We perform a bifurcation analysis of the model to determine the epidemic threshold and derive asymptotic approximations of the endemic equilibrium under two parameter regimes. As well, we perform sensitivity analysis on the results for the endemic equilibrium with respect to network parameters, and find implications for public health interventions that are in line with previous studies.

Chapters 3 and 4 both model processes of social dynamics using adaptive networks, or networks whose edges change dynamically over time. In Chapter 3, we introduce an SEIR model on a heterogeneous, clustered network with random link activation/deletion dynamics. With this framework, we develop realistic mechanisms for social distancing policies using piecewise constant activation/deletion rates for edges in the network. These mechanisms are able to produce rich qualitative behavior and provide insight into what makes for an effective social distancing intervention. Chapter 4 extends this examination of changing social behavior. I introduce a novel dynamical process where random link activation/deletion occurs on a bipartite network where individuals connect to mixing locations and consider its implications for the corresponding unipartite contact network. This new process is analyzed in conjunction with an SIS-type disease spreading on the contact network. Furthermore, I consider the implication for seasonal social dynamics, including how separate sources of seasonality (transmission and social behavior) impact how disease dynamics unfold.